Differentiable Structures on Spheres and the Kervaire Invariant

Abstract

We follow Kervaire Milnor in defining and studying the group G of smooth structures on the sphere S^n. Surgery theory is developed and applied to study the subgroup bP^(n+1) of G. The Pontryagin construction induces a monomorphism p:G/bP^(n+1)->π_n(S)/Im(J), into the cokernel of the stable J-homomorphism. Using surgery theory and the Kervaire invariant the index of p is seen to be 1 unless n=4k+2 and there exist a closed manifold of Kervaire invariant one of dimension n. We also consider Kervaires construction of a piecewise linear manifold admitting no smooth structure.